Purdue UniversityPurdue e-PubsWeldon School of Biomedical Engineering FacultyPublications Weldon School of Biomedical Engineering

1979

Measurements of Young's Modulus of Elasticity ofthe Canine Aorta with UltrasoundD J. Hughes

Charles F. BabbsPurdue University, babbs@purdue.edu

L A. Geddes

J D. Bourland

Follow this and additional works at: http://docs.lib.purdue.edu/bmepubs

Part of the Biomedical Engineering and Bioengineering Commons

This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu foradditional information.

Recommended CitationHughes, D J.; Babbs, Charles F.; Geddes, L A.; and Bourland, J D., "Measurements of Young's Modulus of Elasticity of the CanineAorta with Ultrasound" (1979). Weldon School of Biomedical Engineering Faculty Publications. Paper 52.http://docs.lib.purdue.edu/bmepubs/52

1

ULTRASONIC IMAGING 1, 356-367 (1979)

MEASUREMENTS OF YOUNG'S MODULUS OF

ELASTICITY OF THE CANINE AORTA WITH

ULTRASOUND

D.J. Hughes, C.F. Babbs, L.A. Geddes, and J.D. Bourland

Biomedical Engineering Center

Purdue University

West Lafayette, Indiana, USA.

Abstract

We have developed an ultrasonic technique for determining the dynamic Young's modulus of

elasticity (E) of the canine aorta in vivo. Young's modulus was measured in the descending

thoracic aorta (DTA) and the abdominal aorta (AA) of 12 dogs over a range of mean blood

pressures from 40-200 mmHg. The vessels were excised and dynamic moduli were determined

in vitro postmortem from pressure-volume curves. The data so obtained were compared to the in

vivo values. In vivo and in vitro moduli increased exponentially with mean distending pressure

(P). The equation of best fit for these data was of the form E= E0 exp(aP). Constants E0 and a

depended on the site of measurement (AA or DTA) and upon the particular animal. In vivo and

in vitro moduli were not significantly different in the AA (AA: in vivo E0 = 667 ± 382 mmHg, a

= 0.017 ± 0.004 mmHg-1; in vitro E0 = 888 ± 367, a= 0.016 ± 0.002). However, in vivo moduli

exceeded in vitro moduli in the DTA. (DTA: in vivo E0 = 687 ± 241, a = 0.016 ± 0.004; in vitro

E0 = 349 ± 64, a= 0.018 ± 0.003). The increased stiffness of the DTA compared to the AA in

vivo may be due to the in situ tethering of the aorta to the spine by the parietal pleura.

Key words: Arterial compliance; arterial elasticity; pulse wave velocity

This work was supported by grant #APR-75-1Al89, National Science Foundation, Washington,

D.C., USA.

2

Introduction

The normal mechanical function of the aorta is storage of part of the systolic outflow of the heart

as elastic energy in the aortic wall, which is released to provide flow during diastole. However,

stiffening of arteries occurs with age and may be associated with symptomatic and life

threatening cardiovascular disease. An increase in aortic stiffness increases the work load of the

heart.

The fundamental elastic property of the aortic wall is Young's modulus of elasticity. Young's

modulus is an inherent property of any solid or elastomer, and is defined as the ratio of tensile

stress to tensile strain. Studies of the mechanical/elastic properties of the aorta have been limited

because there have been few methods of measuring the modulus of elasticity of the aorta in vivo.

Studies of excised vessels are of limited validity. This paper describes a new ultrasonic method

for the in vivo measurement of Young's modulus in the intact, living dog and compares in vivo

measurements with in vitro values determined immediately postmortem in these same aortic

segments. By comparing the in vivo and in vitro values, differences between aortic elasticity in

the living animal and in excised vessels can be discerned.

Methods and materials

In the present study, the in vivo Young's modulus of elasticity (E) was calculated from the aortic

pulse wave velocity (Vp), the diameter (d), thickness (t) of the aorta, and the density of blood ().

The mathematical relationship describing these quantities is the Moens-Korteweg equation [1],

d

EtVp

. (1)

Aortic diameter and thickness were measured with a triaxial ultrasound catheter. The density of

blood was determined from the hematocrit [3]. The pulse wave velocity was measured in the

aorta by determining the difference in pressure pulse arrival time with a catheter-tip pressure

transducer at two sites a known distance apart. Due to the small relative diameters of the

catheters with respect to the diameter of the aorta, the presence of the catheters in the vessel was

not expected to modify blood flow. Using this technique, we determined the modulus of

elasticity in the thoracic and abdominal aortae of 12 mongrel dogs. In 8 of the 12 animals,

Young's modulus was measured in segments of both the thoracic and abdominal aortae. In each

of the 12 dogs, after the in vivo measurements were completed, the aortae were excised and

moduli were determined in vitro by a pressure-volume technique.

3

a. In vivo measurements

Twelve dogs anesthetized with sodium pentobarbital (30 mg/kg i.v.) served as subjects. In each

animal, measurements of aortic pulse wave velocity and aortic diameter and thickness were made

over a wide range of mean aortic blood pressure (40-200 mmHg). Blood pressure was elevated

by intravenous injections of phenylephrine hydrochloride (0.1 mg/kg), a short-acting peripheral

vasoconstrictor. Blood pressure was lowered by intravenous doses of sodium nitroprusside (0.1

mg/kg), a short acting peripheral vasodilator. Atropine sulfate (1 mg, i.v.) was given prior to

phenylephrine, and propranolol hydrochloride (1 mg, i.v.) was given prior to nitroprusside to

blunt the action of autonomic reflexes.

A specially designed ultrasonic catheter (developed with the Indianapolis Center for Advanced

Research, Indianapolis, Indiana) was employed to measure aortic diameter and thickness. This

catheter contained three piezoelectric elements spaced equiangularly (120 degrees) around the

catheter tip in a plane perpendicular to its long axis. The catheter (7 French) was 60 cm long. The

piezoelements were excited by a Panametrics (Waltham, Massachusetts) 5050PR pulser, which

also provided preamplification for the echo signals received from the aortic walls. A specially

designed electronic commutator was used to excite each piezoelectric element sequentially and

display the echo patterns on a high-frequency oscilloscope. A pulse repetition frequency of 1

kHz was employed. This ultrasonic catheter system has been described in detail [2]. Figure 1

diagrams the experimental apparatus for the in vivo measurements. Echo patterns on the

oscilloscope were recorded on videotape for later analysis. The videotape recordings allowed

determination of aortic pressures and radii every 33 ms [2].

Figure 1. Diagram of the experimental apparatus for the in vivo measurements.

4

To determine aortic dimensions in vivo, the triaxial ultrasonic catheter was advanced down the

right carotid artery into the desired position in the aorta. In eight of the twelve animals,

measurements were made at two sites, one in the descending thoracic aorta (DTA) distal to the

aortic arch at the mid-thoracic level, and the other in the infrarenal abdominal aorta (AA).

Figure 2 illustrates the technique of measuring aortic diameter from ultrasonic times-of-flight

(tof). Three aortic inner- and outer-wall echoes were displayed on an oscilloscope screen, the

image of which was recorded on video cassette tape for later analysis. By replaying the video

recorded echo signals on a stop-frame video cassette player (Sony V0-2850), ultrasonic times-of-

flight to and from the inner- and outer­ walls of the aorta could be measured. Using previously

determined values for the speed of 10 MHz ultrasound in canine blood [3], and in canine aortic

wall [4], the distances from each piezoelement to the inner- and outer-wall were then computed,

as shown in Figure 2. By applying the polygon law and the law of cosines to an inscribed

triangle whose vertices were the points of intersection of the ultrasonic paths with the circular

vessel walls, the inner- and outer-radii of the aorta were computed. Since only veins collapse to

any significant extent, the assumption of circular aortic section for distending pressures greater

than 40 mmHg should not hinder the application of the polygon toward calculation of vessel

dimensions.

Figure 2. Diagram of catheter location in the aortic lumen, typical A-scan display of

echoes, and the equation relating vessel radius to intravascular distances.

5

To facilitate calculation of mean, systolic and diastolic diameters in the aorta, a blood pressure

channel was displayed by the oscilloscope and recorded on video tape, along with ultrasonic

echoes. Mean inner diameter, Dm, was calculated in a manner analogous to mean blood pressure,

Pm from systolic and diastolic pressures (Ps, Pd) and diameters (Ds, Dd):

dssm PPKPP

(2)

dssm DDKDD .

The parameter K was evaluated from strip chart records of blood pressure. The difference

between mean inner- and outer-wall radii was interpreted as the mean wall thickness, t.

The pulse wave velocity is the remaining quantity needed to calculate Young's modulus using

Eq. (1). Pulse wave velocity was measured using differential pulse arrival times at two sites a

known distance apart. The R-wave of the electrocardiogram was used as the reference time. This

method, described by Patel et al. [5], measures the velocity at which the upstroke of the blood

pressure waveform travels. Pulse arrival was detected by strain gage pressure transducer

mounted at the tip of a 120 cm long #5 French catheter placed adjacent to the ultrasound

catheter-tip transducer.

To calculate pulse wave velocity at the site of ultrasonic measurement, the Millar blood pressure

catheter was translated 5 cm cephalad from the tip of the ultrasonic catheter for a few cardiac

cycles and then moved 5 cm caudad. By using the R-wave of the ECG (lead II) as a timing

reference, measuring the difference in pulse arrival times, T, over the 100 mm excursion gave the

values necessary to compute the pulse wave velocity, Vp :

msT

mm100Vp . (3)

Initial adjacent location of the two catheter tips was verified by echoes received from the blood

pressure catheter with one of the ultrasound transceivers. Postmortem examination of catheter

positions also confirmed adjacency of the two catheter tips. In theory, blood flow velocity must

be subtracted from the calculated pulse wave velocity to obtain the true pulse wave velocity.

However, since the flow velocity is of the order of 0.25 m/s compared to 5 m/s for the pulse

wave velocity [6], this correction was not made in this study. For calculation of in vivo moduli,

values of the pulse wave velocity at specific mean pressures were taken from parabolic

regression functions fitted to the pulse wave velocity data [ 7].

6

b. In vitro measurements

After the in vivo data were obtained, the animal was sacrificed and specimens of the DTA and

AA were excised for determination of the in vitro modulus of elasticity. The excised specimens

coincided with the points of ultrasonic measurement. Before excision, the in situ length, L , was

determined carefully. The specimens were tethered to this length in a room temperature water

bath for determination of Young's modulus by the pressure-volume method shown in Figure 3.

An infusion pump was used to pressurize the vessel with a continuous infusion of saline loaded

with India ink. The ink particles inhibited leakage from the vasa vasorum. The rate of infusion

was constant at 0.5 cm3/s. Transmural pressure (Pt) was recorded with a strain gauge pressure

transducer. Movement of the infusion pump shaft engaged a linear potentiometer to record the

volume infused. Both transmural pressure and infused volume were recorded on an x-y plotter to

produce a pressure-volume curve.

Careful calculation of the internal volume of the vessel (V0) at zero transmural pressure allowed

determination of vessel volume, V, at any pressure from the infused volume. The vessel wall

thickness at zero transmural pressure, t0 , was measured directly by a special tissue micrometer,

by applying Archimedes Principle and knowing vessel dimensions and by using a magnifying

optical gradicule. In each case, the vessel was tethered to the in situ length. Then wall thickness

for any distending volume was calculated, assuming constant cross sectional area of the vessel

wall (Poisson's ratio = 0.5) from the expression

V

Vt)V(t 0

0 . (4)

Young's modulus was calculated from Eq. (5), as described by Posey [6],

L

V

t

2

V

PE

3

. (5)

In this expression, P/V denotes the slope of the P-V curve, t is wall thickness, V is the internal

volume, and L is the in vivo length of the vessel.

7

Figure 3. Diagram of the apparatus used in the in vitro measurements.

Results

Typical in vivo and in vitro values of Young's modulus are represented by the semi-log plot in

Figure 4. The linear relationship obtained suggested an exponential representation of Young's

modulus, E, in terms of mean distending pressure, P. This relationship was described by the

expression

)Paexp(EE 0 . (6)

8

Figure 4. Young's modulus in vivo and in vitro for the thoracic (a) and abdominal (b)

aortae.

9

From semi-log treatment of these data, the constants E and a were determined for each of the

12 dogs. The mean values, E0 and a , are shown in Table 1, along with the ratio of in vivo to in

vitro results. The correlation coefficients obtained for these exponential fits were in excess of

0.95.

Table 1

Discussion

The Moens-Korteweg equation, Eq. (1), was derived in 1878 from the wave equation for

propagation of a pressure impulse within a thin walled, perfectly elastic, cylindrical tube [1].

Details of the derivation were reviewed by Hardung [8]. Measurement of Young's modulus of

the canine aorta in vivo was made on the basis of this relationship. The moduli so evaluated are

necessarily dynamic, that is, they depend on the frequency of application of the distending force.

In this investigation, the distending frequency was the heart rate which ranged from 2.0 to 2.5 Hz

(120-150/min) due to phenylephrine and nitroprusside. Bergel [9] found that the dynamic

modulus was independent of distending frequency for frequencies greater than 2 Hz and less than

10 Hz.

The in vitro measurements of Young's modulus represent values obtained by tethering the aorta

to its in vivo length. This tethering prevents longitudinal extension of the vessel, a natural

phenomenon in vivo, but the introduction of such end effects is minimized by tethering at the in

vivo length [6].

The moduli evaluated in vitro represent dynamic moduli due to the nature of the distension

achieved by employing the infusion pump. By setting the rate of infusion at about 0.5 cm3/s,

10

(yielding a dPt/dT of about 200 mmHg/s), the rate of distension in vitro was equivalent to the in

vivo rate of distension.

Because Young's modulus of elastomeric materials, such as blood vessels, is known to be strain

dependent, the choice of an independent variable for the in vivo measurements was carefully

considered. The circumferential elasticity of vessels is primarily related to radius rather than

pressure [10]. However, since mean blood pressure and vessel radius are intimately related for

the aorta [11], and because mean blood pressure is directly measurable, mean blood pressure was

used as the independent variable for determining the pressure-dependent terms in the Moens-

Korteweg equation.

a. Exponential elasticity

Others have described the nonlinear elastic behavior of blood vessels [6, 8, 10, 12-14]. However,

we found the modulus of elasticity to be exponentially dependent upon mean distending pressure

(Figure 4). Our studies indicate that this nonlinear elastic behavior of the aorta, described by Eq.

(6), applies to both in vivo and in vitro situations. Constants E0 and a were the intercept and

slope, respectively, of the line resulting from a semi-log plot of E vs. P. Therefore, the aorta can

be elastically characterized by E0 and a over the range of mean pressure.

A similar dependence of the dynamic modulus upon distension was obtained in vitro by Hardung

[8]. Subjecting his data to a semi-logarithmic representation yields an exponential equation of

best fit:

)/9.2exp(1022E 5 , (7)

where / is the initial extension of the of thoracic aorta, being the length of the aortic strip

along the direction of extension. The correlation coefficient obtained was 0.975.

Bergel [15] measured an incremental (or static) Young's modulus in the thoracic aorta of a dog

and found a nonlinear dependence with pressure. We have applied an exponential fit to his data

with a resulting equation of best fit

E = 717 (mmHg) exp(0.016 P). (8)

The correlation coefficient for this fit was 0.99. Others have compared measurements of the

static and dynamic modulus of the canine aorta in vivo and in vitro [6,16,17]. Table II

summarizes the results reported here and elsewhere. The range of values reported herein

compare well with the results of these studies.

11

Table II. Comparison of values of Young's modulus reported by several investigators (after [19]).

Our technique of measuring aortic diameter and wall thickness, unlike many previous methods,

was conducted without dissection of the connective tissue surrounding the aorta [2]. In studies of

pressure-diameter relationships in dogs, other investigators disturbed the intact vascular

geometry to enable implantation of transducers for aortic diameter measurement [1, 2, 7].

b. In vivo and in vitro elasticity

From comparison of the ratios given in Table I, it can be seen that the in vivo moduli in the

thoracic aortae were greater than their in vitro counterparts over the same pressure range. The in

vivo and in vitro moduli of the abdominal aortae were approximately the same. We speculated

that this in vivo excess might be attributable to the differences in dynamic moduli reported by

Bergel [15], due to the frequency dependence of the dynamic modulus. The ratio of the dynamic­

to-static modulus for the thoracic aorta increases rapidly from unity at 0 Hz to about 1.07 at 2

Hz. For higher frequencies, this ratio remains essentially constant. Since the ratio of static-to-

dynamic moduli was as high as 3.3 (dog no. 3) and consistently above 1.07 in our experiments,

the differences were not expected to be due to this frequency effect alone.

12

A test of statistical significance of the greater magnitude of the modulus in vivo in the DTA was

made by applying a paired Students t-test to the values of the zero-pressure modulus, E0 , and to

values of Young's modulus calculated in vivo and in vitro at pressures of 75, 100, 125 and 150

mmHg [20]. For the zero-pressure modulus, the mean difference between in vivo and in vitro

moduli was 338 mmHg and this excess was significant at the p < 0.005 level. At 75 mmHg, the

mean excess was 721 mmHg and the significance level was p < 0.01. Similarly, at 100 mmHg,

the mean excess was 1298 mmHg and the level of significance p < 0.005. At 125 mmHg, the

mean excess was 1661 mmHg and the significance level was calculated to be p < 0.01. Finally, at

150 mmHg, the mean excess was 2131 mmHg and it was significant at the p < 0.02 level.

The difference between in vivo and in vitro moduli in the thorax may be attributable to another

factor yet unaccounted for. In our thoracic surgery, we noted that the thoracic aorta was tightly

tethered to the spine by the parietal pleura and intercostal arteries plus surrounding connective

tissue. The abdominal aorta lacks such tethering. This tethering may account for the increased

stiffness in vivo. The indication is for further investigation, repeating the in vivo measurement

with the connective tissue bed and intercostal arteries dissected away.

Summary and conclusions

Young's modulus of elasticity was measured in the canine aorta in vivo, in the intact animal, and

in vitro, in excised vessels of 12 dogs over a mean pressure range of 40-200 mmHg.

Measurements were made by determining the values of the various terms appearing in the

Moens-Korteweg equation. Aortic diameter-to-thickness ratio was determined ultrasonically and

invasively. The pulse wave velocity was determined from pulse arrival times at a catheter-tip

pressure transducer. These same aortae were excised postmortem for evaluation of in vitro

moduli by a standard pressure-volume technique.

Both in vivo and in vitro moduli were found to increase exponentially with mean distending

pressure. These moduli were found to compare very well with previously reported moduli. The

in vivo moduli of the descending thoracic aortae were in excess of the in vitro moduli of the

thorax, perhaps due to the tethering of the DTA to the parietal pleura in vivo.

Acknowledgements

The authors would like to thank Mr. M. Voelz and Mr. N. Fearnot, students at the Biomedical

Engineering Center, for their help in the completion of the experiments. The help and support of

the entire staff and students of the Center was greatly appreciated.

13

References

[1] Korteweg, D. J., Uber die fortpflanzungs-geschwindigkeit des schalles in elastischen

rohren. (On the velocity of transmission of mechanical waves in elastic tubes), Annalen

der Physik und Chemie 5, 525-542 (1878).

[2] Hughes, D. J., Geddes, L. A., Bourland, J. D., and Babbs, C. F., Dynamic Imaging of the

Aorta In Vivo with 10-MHz Ultrasound, in Acoustical Imaging, A. Metherell, ed., Vol. 8,

pp. 699-707 (Plenum Publishing Corp., New York 1979).

[3] Hughes, D. J., Geddes, L. A., Babbs, C. F., Bourland, J. D., and Newhouse, V. L.,

Attenuation and speed of 10-MHz ultrasound in canine blood of various packed cell

volumes at 37 C. Med. & Biol. Engr. & Comp. 17, 619-622 (1979).

[4] Hughes, D.J. and Snyder, B., Speed of 10-MHz sound in aortic wall: Effects of

temperature, storage and formalin soaking, Med. & Biol. Engr. & Comp. 18, 220–222

(1980).

[5] Patel, D. J. and Fry, D.L. The elastic symmetry of arterial segments in dogs, Circ. Res.

24, 1-8 (1969).

[6] Posey, J. A., Jr., Elasticity of arteries. Ph.D. Thesis, Baylor College of Medicine,

Houston, Texas, (1972).

[7] Mood, A.M. and Graybill, F. A., Introduction to the Theory of Statistics (McGraw-Hill,

N.Y., 1963).

[8] Hardung, V., Propagation of Pulse Waves in Viscoelastic Tubing, in Handbook of

Physiology, W. F. Hamilton and P. Dow, eds., Sec. 2, Circulation, Vol. I, pp. 107-135,

(American Physiological Society, Washington, D.C., 1962).

[9] Bergel, D. H., The dynamic elastic properties of the arterial wall. J. Physiol. 156, 458-469

(1961).

[10] Bergel, D. H., The Properties of Blood Vessels, in Biomechanics, Y. C. Fung et al., eds.,

pp. 105-139 (Prentice Hall, Inc., Englewood Cliffs, New Jersey, 1972).

[11] Bergel, D. H., Arterial Visco-Elasticity, in Pulsatile Blood Flow, E.O. Attinger, ed., pp.

275-289 (McGraw­ Hill, New York, 1964).

[12] King, A. L., Pressure-volume tubes with elastomeric walls: siol. ll• 501-508 (1946).

relation for cylindrical human aorta, J. Appl. Physiol 17, 501-508, (1946).

14

[13] Patel, D. J., Janicki, J. S., and Carew, T. E. Static anisotropic elastic properties of the

aorta in living dogs, Circ. Res. 25, 765-779 (1969).

[14] Remington, J. W., Hamilton, W. F., and Dow, P. Some difficulties involved in the

prediction of stroke volume from the pulse wave velocity, Am. J. Physiol. 144 536- 544

(1945).

[15] Bergel, D. H., The static elastic properties of the arterial wall, J. Physiol. 156, 445-457

(1961).

[16] Gow, B. s. and Taylor, M. G. , Measurement of elastic properties of arteries in the living

dog. Circ. Res. 23, 111-122 (1968).

[17] Nickerson, J. L. and Drozic, J.B., Young's modulus and breaking strength of body tissues.

Technical Document No. AMRL-TDR-64-23, U.S. Air Force 6570, Aerospace Medical

Research Lab., pp. 1-11 (1964).

[18] Barnett, G. 0., Hallos, A. J. and Shapiro, A. Relationship of aortic pressure and diameter

in the dog, J. Appl. Physiol. 16, 545-548 (1961).

[19] Bertram, C. D., Ultrasonic transit-time system for arterial diameter measurement, Med. &

Biol. Engr. & Comp. 11, 489-499 (1977).

[20] Ostle, B., Statistics in Research , Iowa State University Press, Des Moines, Iowa, (1963).